AP CALCULUS BC SUMMER ASSIGNMENT

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1 AP CALCULUS BC SUMMER ASSIGNMENT Dear BC Calculus Student, Congratulations on your wisdom in taking the BC course! We know you will find it rewarding and a great way to spend your junior/senior year. This course is primarily concerned with developing your understanding of the concepts of calculus and providing eperience with its methods and applications. The course emphasizes a multi-representational approach to calculus, with concepts, results, and problems being epressed graphically, numerically, analytically, and verbally. In order to be successful in this course you need the proper foundation (i.e. knowledge of algebra, geometry, trigonometry, analytic geometry, and elementary functions). You will have to be very familiar with the basic families of functions, and all of their representations, in order to be successful in your study of calculus. The concept of functions underlies everything that calculus considers. You will also need to be able to carry out certain computational tasks (i.e. algebra skills) with efficiency and accuracy if you are going to be successful in calculus. These include manipulations of functional symbolism, solving algebraic equations involving the functions mentioned above, interpreting numerical values given by formulas, graphs, and tables, using and manipulating data, and knowing how and when to use your calculator. This is a rigorous college course. The curriculum and pace of the course is intense and is designed to prepare students for the AP eam. Since this is a college class you can epect to spend approimately - hours completing homework or studying for every hour that you are in class learning. Each test and quiz that is given is cumulative and will be graded as per the College Board guidelines. Therefore, this course will be challenging and demanding. Net year, the AP eam will be on Tuesday, May 4 th. Circle this day on your calendar. The College Board requires the use of a graphing calculator on the eam. If you do not own a graphing calculator, we strongly encourage you to purchase one this summer. Spend some time familiarizing yourself with the capabilities of your calculator. The school does have class sets of TI-84 s that will be available in class but not outside of class. Complete this packet over the summer. Please be advised that on the third day of class you will be required to turn in the packet (0 pts) and take a quiz (0 pts) on the information within the first two weeks of school. This quiz will cover the following concepts and skills: its, all differentiation methods and application of derivatives. These were covered in the first three chapters of Calculus AB/Math Analysis. Moreover, you will be assessed on this material throughout the entire year, so it is in your best interest to review the sample questions provided on the subsequent pages, and prepare yourself prior to the first day of school. Be advised that these are the skills that we epect you to possess prior to the first day of school. Therefore, if you are unable to answer some of these questions, we suggest that you start studying! Feel free to contact your teachers with any questions or concerns that you or your parents may have. Our addresses are included below. Have a restful summer and be ready to talk math again in August. Sincerely, Mrs. Maria Quinn James maria.james@lcps.org

2 . Find an equation for the line that contains the points (, -) and (6, 9).. Find the value of y for which the line through A and B has the given slope m: A(-, ), B(4, y), m.. Find an equation for the line that contains the coordinate (, ) and is perpendicular to the line 6 y. For questions 4-9, let and g( ). 4. ( f g)( ). ( g f )(6) 6. g ( ) 7. ( g f )( ) ( f g)() 0. Algebraically find the inverse of y. For questions -9, simplify each epression completely... ln e. ln 4 7 ln e. log 8 6. e ln

3 7. 4y y ( ) ( ( ) ) 0. Rewrite ln( ) ln( ) 6ln as a single logarithmic epression. t. Solve for t: (.04). Solve for : log log ( 4). Solve for : Solve for : ln() 6. Evaluate log to the nearest thousandth. 6. Solve: 0 7. Solve: Without using a calculator, find the eact value of cos 7 cos. Justify your answer. For questions 9-4, find the eact values of each trigonometric function sin 0. csc. 6 cos. sec. tan 4. cot( )

4 . Simplify (csc( ) tan( ))sin( )cos( ) 6. List the three Pythagorean Identities. 7. List the double angle formulas a) sin = b) cos = 8. List the sum and difference formulas. a) cos( ) b) sin( ) sin cos 9. Prove that csc cos sin 40. Prove that (sin cos ) sin 4. For the solution of the equation sin sin for Find the domain for k ( ) Determine all points of intersection for y 4 and y. 44. Find the points of intersection in the graphs of y and y Use a graphing calculator to approimate all of the function s real zeros. Round your results to 6 decimal places. 4

5 46. Algebraically determine whether the function is even, odd, or neither: a. 7 b. 4 c If, describe in words what the following would do the graph of f (). a. f () 4 b. f ( 4) c. f ( ) d. 48. Let and g ( ). Which of the following are true? I. g ( ) f ( ) for all values of. II. ( f g)( ) for all values of. III. The function is one-to-one 49. For the function below, give the zeros (if non eist, write none), domain, range, VA s, HA s, and/or points of discontinuity (holes-as ordered pairs)if any eist. Also sketch the function s graph. For questions 0-, graph each function on the attached graph paper. Give its domain and range. 0. y e. y =. y sin. y 4. f (),, 0 0.,,, (, ) (,) (, )

6 Use the graph below to answer questions f () 78. f ()

7 79. Find the 80. Find the ( 4) 8. Find the 8. Find the Find the 84. Find the t 6t t ( t)(t ) 8. Find the cos 86. Find the Find the Find the 89. Find the Find the slope of the tangent line to the graph of g ( ) 4 at the point (, -). 9. Find an equation of the tangent line to the hyperbola y at the point (, ). 9. Find an equation of the tangent line to the graph of 8 9 a the point (,-6). For questions 9-9, find the -values (if any) at which f is not continuous. Which of the discontinuities are removable. 9. ( ) 94. cos 9. f 0

8 For questions 96-6, find the derivative of each given function y e g ( ) f ( ) e tan (8 ) e 06. y sin cos 07. y cos 08. cos tan 09. cot sec g ( ) ( ). y 7. 4 y ( ) ( ). y ln( ) 4. Find y of y log 0.. If e, find the value of f '() 6. If, find ( ),, f and f ()

9 dy For questions 7-8, find by implicit differentiation. d 7. y 8. y y 8 9. If a billiard ball is dropped from a height of 00 feet, its height s at time t is given by the position function: s 6t 00 where s is measured in feet and t is measured in seconds. Find the average velocity of each the following time intervals. a) [, ] b) [,.] c) [,.] 0. A particle moves according to a law of motion s f ( t) t t 6t measured in seconds and s in meters. a) Find the velocity at time t. b) What is the velocity after s? c) When is the particle at rest? d) Find the total distance traveled during the first 8 s. e) Find the acceleration at time t. f) Find the acceleration after s., t 0, where t is

10 . The length of a rectangle is decreasing by inches per second and the width is increasing by inches per second. When the length is 0 inches and the width is 6 inches, how fast is the (a) perimeter and (b) area changing?. A rectangle has its vertices on the -ais, the y-ais, the origin, and the graph of y 4 in the first quadrant. Find the maimum possible area for such a rectangle. Justify your answer

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